TITLE: Trying to understand Hilbert Spaces...
QUESTION [6 upvotes]: I am trying to get a hold on Hilbert Spaces, but I am having difficulties combinging various definitions. 
I have looked it up on wikipedia and wolfram, there it states something like "A Hilbert space $H$ is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product.".
In Croom's Principles of Topology it says "Hilbert space $H$ consists of all infinite sequences $x=(x_1,x_2,\dots,x_n,\dots )$ for which each coordinate $x_i$ is a real number and for which the sum of all $x_{i}^{2}$ converges to a finite limit $\dots$
(Im a little confused) Simply put... What is a Hilbert Space?
I am trying to get a Topologists point of view on the matter. I read that $H$ has some useful properties that come in handy when proving some theorema, so..

REPLY [3 votes]: Consider a vector space over $\mathbb{R}$ (or $\mathbb{C}$). This can be made into an inner product space by specifying an inner product $\langle*,*\rangle,$ which takes two vectors and returns an element of $\mathbb{R}$ (or $\mathbb{C}$). Finally, a Hilbert space is a particularly nice kind of inner product space. Technically, a Hilbert Space is an inner product space that is complete with respect to the norm $ \|x\| = \langle x,x\rangle^{1/2}$. This just means it needs to be complete with respect to the metric $d(x,y) = \langle x-y,x-y\rangle^{1/2}$. Note that we have defined the metric such that $d(x,y)=\|x-y\|.$
I don't know the details about the useful properties of Hilbert space, but broadly speaking, everyone agrees that Hilbert Space is the best. Everyone loves Hilbert space.